\(\def\N{\mathbb{N}}\) \(\def\Z{\mathbb{Z}}\) \(\def\Q{\mathbb{Q}}\) \(\def\R{\mathbb{R}}\) \(\def\C{\mathbb{C}}\) \(\def\H{\mathbb{H}}\) \(\def\6{\partial}\) \(\DeclareMathOperator\Res{Res}\) \(\DeclareMathOperator\M{M}\) \(\DeclareMathOperator\ord{ord}\) \(\DeclareMathOperator\const{const}\) \(\DeclareMathOperator{\arccosh}{arccosh}\) \(\DeclareMathOperator{\arcsinh}{arcsinh}\) \(\DeclareMathOperator\id{id}\) \(\DeclareMathOperator\rk{rk}\) \(\DeclareMathOperator\tr{tr}\) \(\def\pt{\mathrm{pt}}\) \(\DeclareMathOperator\colim{colim}\) \(\DeclareMathOperator\Hom{Hom}\) \(\DeclareMathOperator\End{End}\) \(\DeclareMathOperator\Aut{Aut}\) \(\let\Im\relax\DeclareMathOperator\Im{Im}\) \(\let\Re\relax\DeclareMathOperator\Re{Re}\) \(\DeclareMathOperator\Ker{Ker}\) \(\DeclareMathOperator\Coker{Coker}\) \(\DeclareMathOperator\Map{Map}\) \(\def\GL{\mathrm{GL}}\) \(\def\SL{\mathrm{SL}}\) \(\def\O{\mathrm{O}}\) \(\def\SO{\mathrm{SO}}\) \(\def\Spin{\mathrm{Spin}}\) \(\def\U{\mathrm{U}}\) \(\def\SU{\mathrm{SU}}\) \(\def\g{{\mathfrak g}}\) \(\def\h{{\mathfrak h}}\) \(\def\gl{{\mathfrak{gl}}}\) \(\def\sl{{\mathfrak{sl}}}\) \(\def\sp{{\mathfrak{sp}}}\) \(\def\so{{\mathfrak{so}}}\) \(\def\spin{{\mathfrak{spin}}}\) \(\def\u{{\mathfrak u}}\) \(\def\su{{\mathfrak{su}}}\) \(\def\cA{\mathcal{A}}\) \(\def\cB{\mathcal{B}}\) \(\def\cC{\mathcal{C}}\) \(\def\cD{\mathcal{D}}\) \(\def\cE{\mathcal{E}}\) \(\def\cF{\mathcal{F}}\) \(\def\cG{\mathcal{G}}\) \(\def\cH{\mathcal{H}}\) \(\def\cI{\mathcal{I}}\) \(\def\cJ{\mathcal{J}}\) \(\def\cK{\mathcal{K}}\) \(\def\cL{\mathcal{L}}\) \(\def\cM{\mathcal{M}}\) \(\def\cN{\mathcal{N}}\) \(\def\cO{\mathcal{O}}\) \(\def\cP{\mathcal{P}}\) \(\def\cQ{\mathcal{Q}}\) \(\def\cR{\mathcal{R}}\) \(\def\cS{\mathcal{S}}\) \(\def\cT{\mathcal{T}}\) \(\def\cU{\mathcal{U}}\) \(\def\cV{\mathcal{V}}\) \(\def\cW{\mathcal{W}}\) \(\def\cX{\mathcal{X}}\) \(\def\cY{\mathcal{Y}}\) \(\def\cZ{\mathcal{Z}}\) \(\def\al{\alpha}\) \(\def\be{\beta}\) \(\def\ga{\gamma}\) \(\def\de{\delta}\) \(\def\ep{\epsilon}\) \(\def\ze{\zeta}\) \(\def\th{\theta}\) \(\def\io{\iota}\) \(\def\ka{\kappa}\) \(\def\la{\lambda}\) \(\def\si{\sigma}\) \(\def\up{\upsilon}\) \(\def\vp{\varphi}\) \(\def\om{\omega}\) \(\def\De{\Delta}\) \(\def\Ka{{\rm K}}\) \(\def\La{\Lambda}\) \(\def\Om{\Omega}\) \(\def\Ga{\Gamma}\) \(\def\Si{\Sigma}\) \(\def\Th{\Theta}\) \(\def\Up{\Upsilon}\) \(\def\Chi{{\rm X}}\) \(\def\Tau{{T}}\) \(\def\Nu{{\rm N}}\) \(\def\op{\oplus}\) \(\def\ot{\otimes}\) \(\def\t{\times}\) \(\def\bt{\boxtimes}\) \(\def\bu{\bullet}\) \(\def\iy{\infty}\) \(\def\longra{\longrightarrow}\) \(\def\an#1{\langle #1 \rangle}\) \(\def\ban#1{\bigl\langle #1 \bigr\rangle}\) \(\def\llbracket{{\normalsize\unicode{x27E6}}} \def\rrbracket{{\normalsize\unicode{x27E7}}} \) \(\def\lb{\llbracket}\) \(\def\rb{\rrbracket}\) \(\def\ul{\underline}\) \(\def\ol{\overline}\)

5  Power Series

Definition 5.1 A formal power series is an expression of the form

\[P=\sum_{n=0}^\iy a_nT^n \tag{5.1}\]

with coefficents \(a_n\in\C.\) As no convergence is required, this is really just a sequence \((a_n)_{n\in\N}\) of complex numbers.

We call \(a_0\) the constant term of \(P.\) The order of \(P\) is the smallest \(n\in\N\) such that \(a_n\neq0.\)

Example 5.1  

Covered in lectures. Check back once the chapter is concluded.






Definition 5.2 Let \(\C \llbracket T\rrbracket\) be the set of all formal power series. Define the addition and multiplication of \(P=\sum_{n=0}^\iy a_nT^n,\) \(Q=\sum_{n=0}^\iy b_nT^n\in\C\llbracket T\rrbracket\) by \[P + Q=\sum_{n=0}^\iy(a_n+b_n)T^n, \tag{5.2}\] \[PQ=\sum_{n=0}^\iy c_nT^n, c_n=\sum_{i+j=n} a_ib_j. \tag{5.3}\]

Proposition 5.1  

  1. These operations make \(\C\llbracket T\rrbracket\) a commutative ring with unit \(1,\) the series with constant term \(1\) and all higher coefficients zero.
  2. \(P\in\C\llbracket T\rrbracket\) has a multiplicative inverse \(\iff\) the constant term of \(P\) is non-zero.

Proof.

Covered in lectures. Check back once the chapter is concluded.



















Covered in lectures. Check back once the chapter is concluded.






Example 5.2  

Covered in lectures. Check back once the chapter is concluded.






Definition 5.3 The domain \(D(P)\) of a formal power series Equation 5.1 is the set of all \(z\in\C\) such that the series of complex numbers \[P(z)=\sum_{n=0}^\iy a_nz^n,\]

obtained by substituting \(T\) by \(z,\) converges. We obtain a complex function \[D(P)\longra\C, z\longmapsto P(z).\]

More generally, fix a center \(z_0\in\C.\) We then have a complex function

\[D(P,z_0)\coloneqq z_0+D(P)\longra\C, z\longmapsto P(z-z_0)=\sum_{n=0}^\iy a_n(z-z_0)^n \tag{5.4}\]

which differs from \(P(z)\) only by a translation.

Example 5.3  

Covered in lectures. Check back once the chapter is concluded.









Recall the following concept from analysis.

Definition 5.4 For a complex function \(f\colon D\to\C,\) the uniform norm is

\[\|f\|_{\iy,D}=\sup_{z\in D} |f(z)|. \tag{5.5}\]

A sequence of functions \((f_n)_{n\in\N}\) is uniformly convergent to \(f\) on \(D\) if \[\|f-f_n\|_{\iy,D}\to0 \text{ as } n\to\iy.\]

A series \(\sum_{n=0}^\iy f_n(z)\) is a uniformly convergent series on \(D\) if the sequence of partial sums \(P_n(z)=\sum_{k=0}^nf_k(z)\) converges uniformly.

Uniform convergence on a subset \(C\subset D\) refers to the uniform convergence of the functions restricted to \(C.\)

Theorem 5.1 Every formal power series \(P\) has a unique radius of convergence \(0\leqslant\rho\leqslant+\iy\) such that for every center \(z_0\)

\[D_\rho(z_0)\subset D(P,z_0)\subset\ol{D}_\rho(z_0). \tag{5.6}\]

In fact,

\[\rho=\sup\left\{r\geqslant0\ \middle|\ (|a_k|r^k)_{k\in\N}\text{ is a bounded sequence}\right\}. \tag{5.7}\]

Moreover, \(P(z-z_0)\) converges absolutely and uniformly on every smaller disk \(D_r(z_0)\) with \(0<r<\rho\) and diverges at every point \(z\in\C\) with \(|z-z_0|>\rho.\)

Proof.

Covered in lectures. Check back once the chapter is concluded.



















Remark 5.2. The domain \(D(P)\) of a formal power series is roughly a disk of radius \(\rho,\) with uncertain behavior on the boundary circle \(|z|=\rho.\) Determining the behavior on the boundary can be very difficult. On the other hand, the radius of convergence is easy to compute. If you find \(z\in\C\) for which \(P(z)\) converges (for example, using the ratio or the root test), you can deduce \(|z|<\rho.\) If \(P(z)\) diverges at \(z\in\C,\) you know \(|z|\geqslant\rho.\)

Example 5.4 (optional)  

Covered in lectures. Check back once the chapter is concluded.
























Corollary 5.1 The restriction of the complex function \(f(z)=P(z-z_0)\) defined in Equation 5.4 to \(D_\rho(z_0)\subset D(P,z_0)\) is continuous.

Proof.

Covered in lectures. Check back once the chapter is concluded.



















Remark 5.2. Continuity need not hold on \(D(P,z_0)\) (Sierpinski, 1916).

Example 5.5  

Covered in lectures. Check back once the chapter is concluded.









When the radius of convergence is positive, we drop the word ‘formal’ and simply speak of a (convergent) power series.

Example 5.6  

Covered in lectures. Check back once the chapter is concluded.



















Example 5.7 (optional)  

Covered in lectures. Check back once the chapter is concluded.



















Theorem 5.2 Let \(P\) be a formal power series. Assume the radius of convergence \(\rho>0\) is positive. Then for each center \(z_0\in\C\) the function \[P\colon D_\rho(z_0)\longra\C, z\longmapsto P(z)=\sum_{n=0}^\iy a_n(z-z_0)^n \tag{5.8}\]

is holomorphic with derivative given by termwise differentiation,

\[P'(z)=\sum_{n=1}^\iy na_n(z-z_0)^{n-1}. \tag{5.9}\]

Hence \(P\) is infinitely complex differentiable, by induction.

Proof.

Covered in lectures. Check back once the chapter is concluded.



















The identity theorem for power series is the following result.

Corollary 5.2 Let \(P=\sum_{n=0}^\iy a_n z^n,\) \(Q=\sum_{n=0}^\iy b_n z^n\) be formal power series with positive radius of convergence. Let \(0\neq z_\ell\in D(P)\cap D(Q)\) be a null sequence, \(\lim_{\ell\to\iy} z_\ell=0,\) of non-zero complex numbers in the common domain. If \(P(z_\ell)=Q(z_\ell)\) for all \(\ell,\) then all coefficients \(a_n=b_n\) agree.

Proof.

Covered in lectures. Check back once the chapter is concluded.



















Questions for further discussion

  • Give precise statements of the comparison test, the ratio test and root test. Recall how the ratio test is proven by comparison with the geometric series.

  • Find a power series \(P\) with \(D(P)=\ol{D}_1(0)\setminus\{\pm1,\pm i\}.\)

  • Explain why ‘is a bounded sequence’ in Equation 5.7 can be replaced by ‘is a null sequence’.

5.1 Exercises

Exercise 5.1

Find the radius of convergence for the following power series centered at the origin.

  1. \(\sum_{n=0}^\iy\frac{z^n}{n^3}\),
  2. \(\sum_{n=0}^\iy z^{3n}\),
  3. \(\sum_{n=0}^\iy\frac{z^n}{n^n}.\)

Hint: Recall the ratio and root tests for series of complex numbers

Exercise 5.2

Treating \(e^z,\) \(\sin(z),\) \(\cos(z)\) as formal power series with their usual Taylor expansion, find the terms of order \(\leq3\) of the following power series:

  1. \(e^z\sin(z)\),
  2. \(\sin(z)\cos(z)\),
  3. \(1/\cos(z)\)
Exercise 5.3

Let \(P=\sum_{n=0}^\iy a_n z^n,\) \(Q=\sum_{n=0}^\iy b_n z^n\) be power series with positive radii of convergence \(\rho_P, \rho_Q>0.\) Show that:

  1. \(P+Q=\sum_{n=0}^\iy (a_n+b_n) z^n\) has radius of convergence \(\rho\geqslant\min(\rho_P,\rho_Q).\)
  2. \(PQ=\sum_{n=0}^\iy\left(\sum_{i+j=n}a_ib_j\right)z^n\) has radius of convergence \(\rho\geqslant\min(\rho_P,\rho_Q).\)
Exercise 5.4

Find a solution to the non-linear differential equation \[f'(z)+f(z)^2=0,\qquad f(0)=1\] on a disk centered at \(z_0=0\) by making the ansatz \(f(z)=\sum_{n=0}^\iy a_nz^n,\) inductively determining the coefficients \(a_n,\) and finding the radius of convergence.

Exercise 5.5

Let \(P(z)=\sum_{n=0}^\iy a_n z^n\) be a power series centered at \(z_0=0\) and assume that the radius of convergence \(\rho>0\) is positive. Suppose that \(P(z)\in\R\) for all \(z\in D_\rho(0)\cap\R.\) Prove that all coefficients \(a_n,\) \(n\in\N,\) must be real numbers. Deduce that \(\ol{P(z)}=P(\ol{z})\) for all \(z\in D_\rho(0).\)

Exercise 5.6

Prove that the ring of formal power series \(\C\llbracket T\rrbracket\) is an integral domain. In other words, show that \(PQ=0\implies P=0\) or \(Q=0.\)

Exercise 5.7

The binomial coefficient of a complex number \(\al\neq0\) and \(k\in\N\) is defined as \[\binom{\al}{k}= \begin{cases} \frac{\al(\al-1)\cdots(\al-k+1)}{k!} &\text{if }k>0,\\ 1 &\text{if } k=0. \end{cases}\] The is the formal power series \[B_\al = \sum_{k=0}^\iy \binom{\al}{k}T^k.\]

  1. Determine the radius of convergence of \(B_\al.\)
  2. Prove that if \(\al\in\N,\) then \(B_\al(z)=(z+1)^\al.\)
  3. Show the generalized Vandermonde identity for \(\al,\be,\al+\be\in\C^\t,\) \[\sum_{k=0}^n\binom{\al}{k}\binom{\be}{n-k}=\binom{\al+\be}{n}.\tag{$\star$}\]
  4. Prove that for \(\al=1/k\) the complex function \(B_\al(z)\) satisfies \(B_\al(z)^k=z+1\) on its domain. Hence \(B_\al(z)\) is a \(k\)-th root of the function \(z+1.\)
Exercise 5.8

For \(a>0\) and \(z\in\C\) define \(a^z=\exp(\log(a)z).\) Show that:

  1. \(a^zb^z=(ab)^z\) for all \(a,b>0,\) \(z\in\C\)
  2. \(a^za^w=a^{z+w}\) for all \(a>0,\) \(z,w\in\C\)
  3. \(|a^z|=a^{\Re(z)}\) for all \(a>0,\) \(z\in\C\)

The is defined as \[\ze(z)=\sum_{n=1}^\iy\frac{1}{n^z}\qquad\text{for $\Re(z)>1.$}\tag{$\star$}\]

  1. Prove that the series (\(\star\)) converges absolutely for all \(z\in\C\) with \(\Re(z)>1\) and uniformly on every subset \(S_\de=\{z\in\C\mid\Re(z)>1+\de\}\) with \(\de>0.\)